st: re: df adjustment for robust s.e. with xtreg, fe

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In the case of
webuse grunfeld
areg mvalue kstock invest, absorb(company) clu(company)
which should be equivalent to estimating via
xtreg mvalue kstock invest, fe) clu(company)
viewsource areg.ado tells you that
eret scalar rmse = `=sqrt(e(rss)/(e(N)-e(df_a)-e(df_m)-1))'
where df_a is one fewer than the number of clusters, and df_m is the number of slopes.
The areg estimate has e(rmse) = 268.73 whereas the xtreg,fe estimate has e(rmse) = 262.52295. If you take the ratio of those two numbers and square it, and multiply it by the xtreg VCE, you get the areg VCE, so the difference in SEs is explained by this scalar factor.
I think the issue here is model degrees of freedom, e(df_m). Usually model DF in a regression context is the number of slope parameters. Without the cluster option on the commands above, xtreg,fe reports e(df_m) = 11 and areg reports e(df_m) = 2, for a difference of 9: the number of units in the panel minus one, which is consistent with Stata's notion of reporting a constant term for the fixed effects, and suppressing the other fixed-effect coefficients, which would be 9 in number.
But when you turn on the cluster option, areg continues to assert e(df_m) = 2, but xtreg,fe asserts e(df_m) = 1. That one DF is not enough to cause the difference between the two e(rmse) estimates. However, you can reproduce the xtreg,fe RMS Error estimate of 262.52295 by doing sqrt(e(rss)/197). I can't see why the RMS Error from FE should be calculated with (200-3) = 197 DF.
So I think some explanation is necessary. I see no reason, conceptually, why xtreg,fe with small-sample statistics should not be exactly equivalent to areg.
Kit Baum | Boston College Economics & DIW Berlin | http://ideas.repec.org/e/pba1.html
An Introduction to Stata Programming | http://www.stata-press.com/books/isp.html
An Introduction to Modern Econometrics Using Stata | http://www.stata-press.com/books/imeus.html
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